We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula for the determinant is replaced by the number of crossings in the Brauer diagram. This interpretation naturally explains why the determinant of an even antisymmetric matrix is the square of a Pfaffian.

We define a de Bruijn process with parameters n and L as a certain continuous-time Markov chain on the de Bruijn graph with words of length L over an n-letter alphabet as vertices. We determine explicitly its steady state distribution and its characteristic polynomial, which turns out to decompose into linear factors. In addition, we examine the stationary state of two specializations in detail. In the first one, the de Bruijn-Bernoulli process, this is a product measure. In the second one, the Skin-deep de Bruin process, the distribution has constant density but nontrivial correlation functions. The two point correlation function is determined using generating function techniques.

The refined enumeration of alternating sign matrices (ASMs) of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study, starting with the Refined Alternating Sign Matrix Conjecture of Mills-Robbins-Rumsey, its proof by Zeilberger, and more recent work on doubly-refined and triply-refined enumeration by several authors. In this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the "top-left-bottom" (triply-refined) and "top-left-bottom-right" (quadruply-refined) enumerations. The latter case solves the problem of computing the full boundary correlation function for ASMs. The enumeration formulas are proved by deriving new representations, which are of independent interest, for the partition function of the square ice model with domain wall boundary conditions at the "combinatorial point" 2{\pi}/3.

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling.

We demonstrate a natural bijection between a subclass of alternating sign matrices (ASMs) defined by a condition on the corresponding monotone triangle which we call the gapless condition and a subclass of totally symmetric self-complementary plane partitions defined by a similar condition on the corresponding fundamental domains or Magog triangles. We prove that, when restricted to permutations, this class of ASMs reduces to 312-avoiding permutations. This leads us to generalize pattern avoidance on permutations to a family of words associated to ASMs, which we call Gog words. We translate the gapless condition on monotone trangles into a pattern avoidance-like condition on Gog words associated. We estimate the number of gapless monotone triangles using a bijection with p-branchings.

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids.

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset. This gives rise to a strongly connected graph on L. In earlier work (arXiv:1205.7074), we studied promotion-based Markov chains on these linear extensions which generalizes results on the Tsetlin library. We used the theory of R-trivial monoids in an essential way to obtain explicitly the eigenvalues of the transition matrix in general when the poset is a rooted forest. We first survey these results and then present explicit bounds on the mixing time and conjecture eigenvalue formulas for more general posets. We also present a generalization of promotion to arbitrary subsets of the symmetric group.

We define two general classes of nonabelian sandpile models on directed trees (or arborescences) as models of nonequilibrium statistical phenomena. These models have the property that sand grains can enter only through specified reservoirs, unlike the well-known abelian sandpile model. In the Trickle-down sandpile model, sand grains are allowed to move one at a time. For this model, we show that the stationary distribution is of product form. In the Landslide sandpile model, all the grains at a vertex topple at once, and here we prove formulas for all eigenvalues, their multiplicities, and the rate of convergence to stationarity. The proofs use wreath products and the representation theory of monoids.

We develop a general theory of Markov chains realizable as random walks on $\mathscr R$-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via M\"obius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.